10. Simulation/Regression Pricing Schemes in Diffusive Setups

We reviewed simulation and deterministic pricing schemes in Chaps. 6–8. Analogies and differences between simulation and deterministic pricing schemes are most clearly visible in the context of pricing by simulation claims with early exercise features (American and/or cancelable claims). Early exercisable claims can be priced by hybrid “nonlinear Monte Carlo” pricing schemes in which dynamic programming equations, similar to those used in deterministic schemes, are implemented on stochastically generated meshes. Such hybrid schemes are the topics of Chaps. 10 and 11, in diffusion and in pure jump setups, respectively.

In Chap. 10 we deal with the issue of pricing convertible bonds numerically, by simulation. A convertible bond can be regarded as a coupon-paying and callable American option. Moreover, call times are subject to constraints, known as call protections, preventing the issuer from calling the bond at certain sub-periods of time. The nature of the call protection may be very path-dependent, leading to high-dimensional nonlinear pricing problems. Deterministic pricing schemes are then ruled out by the curse of dimensionality, and simulation methods are the only viable alternative. The numerical results of this chapter illustrate the good performances of the simulation regression scheme for pricing convertible bonds with highly path-dependent call protection. More generally, this chapter is an illustration of the power of simulation approaches, which automatically select and loop over the most likely future states of a potentially high-dimensional, but also often very degenerate, factor process X, given an initial condition X ₀=x. By contrast, deterministic schemes loop over all the possible states of X _i regardless of their likelihood. Simulation schemes thus compute “where there is light”. By contrast, deterministic schemes compute everywhere, also in “obscure”, useless regions of the state space. In the context of path-dependent payoffs with a high-dimensional but very degenerate factor process, a simulation scheme thus exploits the degeneracy of a factor process, whereas this path-dependence makes the deterministic scheme impractical.